An approximation scheme for the effective Hamiltonian and applications

We introduce an approximation for first order Hamilton-Jacobi equations with a convex Hamiltonian periodic in the space variable. To this end we use a first order semi-Lagrangian scheme to compute a solution of the so-called cell problem and the related effective Hamiltonian. The scheme can also be interpreted as a discrete version of the Lax-Oleinik representation formula for the exact viscosity solution of the time-dependent problem. The information included in the solutions of the cell problem and in the effective Hamiltonian is exploited for the approximation of the Aubry set. We prove some properties of the scheme and illustrate the effectiveness of our approximation by several tests in dimension 1 and 2.